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Article published in
International Journal of Mechanical Sciences, Vol. 231 (2022) 107572
https://doi.org/10.1016/j.ijmecsci.2022.107572
FFT-based Inverse Homogenization for Cellular
Material Design
Zeyao Chen1, Baisheng Wu1,*, Yi Min Xie2, Xian Wu3 and Shiwei Zhou2,*
1School of Electro-Mechanical Engineering, Guangdong University of Technology,
Guangzhou 510006, China
2Centre for Innovative Structures and Materials, School of Engineering, RMIT
University, GPO Box 2476, Melbourne 3001, Australia.
3School of Automotive Studies, Tongji University, Shanghai 201804, China
Abstract
Besides constituting components, the properties of composites are highly relevant
to their microstructures. The work proposed a fast Fourier transform (FFT)-based
inverse homogenization method implemented by the bi-directional evolutionary
structural optimization (BESO) technique to explore the vast potential of cellular
materials. The periodic boundary condition of self-repeated representative volume
elements can be naturally satisfied in the FFT-based homogenization scheme. The
objective function of the optimization problem is the specific moduli or the quadratic
difference between the effective value and the target, which are obtained in terms of
mutual strain energies. Its sensitivity to the design variable, namely the elemental
density, is derived from the adjoint variable method and used as the criterion to remove
or add material in local elements. Numerical examples show that the proposed method
generates a series of architected cellular materials with maximum modulus, negative
Poisson’s ratio, and specific elasticity tensor. FFT-based homogenization in the method
demands less memory usage but has high efficiency. Thus, it can achieve topology
optimization of unit cell with one million hexahedral elements. This approach
contributes to the extended application of FFT-based homogenization and can guide the
microstructure design of mechanical metamaterials.
Keywords: FFT-based inverse homogenization; Bi-directional evolutionary structural
optimization; Computational material design
-------------------------------------------------------------------------
*E-mail: wubs@gdut.edu.cn (B. Wu)
*E-mail: shiwei.zhou@rmit.edu.au (S. Zhou)
2
Highlights:
► An FFT-based inverse homogenization approach is proposed for efficiently
designing cellular materials.
► Rotating architected materials with negative Poisson’s ratio are generated.
► Architected cellular materials with maximum stiffness and isotropy are sought out.
► The inverse homogenization method for scaffold design with specified mechanical
performance is validated.
1. Introduction
Nowadays, architected cellular materials, a kind of rationally designed artificial
materials consisting of periodically distributed microstructures, are of great interest in
various engineering fields e.g., aerospace, automotive, bio-mechanical industries, or
energetic and chemical sectors, thanks to their tailored performances [1]. The superior
mechanical or physical performances of cellular materials, including ultra-stiffness [2],
negative Poisson’s ratio [3], extreme thermal conductivity [4, 5], high energy
absorption rate [6], low-frequency bandgap [7], and high acoustic absorption capability
[8], exceed the ones of their constituent materials. Especially, the high stiffness-to-
weight or unconventional mechanical property of architected cellular materials has
attracted much attention from mechanical science [9]. Scientists attribute such a
significant performance enhancement to their elaborately designed microstructures,
which also form the foundation of the multi-scale design of cellular materials [10].
However, the traditional design of microstructures relies heavily on intuition or analogy
to existing structures and, therefore, cannot satisfy the demands of advanced cellular
materials with unconventional properties [11]. Advanced manufacturing processes
provide a tremendous opportunity to fabricate materials with precisely defined
architectures, provoking this new microstructural design challenge [12].
The first step in designing cellular materials is to extract their effective properties.
Various approximate methods based on Eshelby's inclusion theory [13], including the
self-consistent method [14], the generalized self-consistent method [15], and Mori-
Tanaka method [16], etc., were developed for single ellipsoidal inclusion problem of
two-phase material. These methods proposed some simple functions to reveal the
dependence of the effective behaviors on the microstructure [17]. However, they are
not suitable for periodic materials with complex microstructures. Some earliest
computational homogenization studies used finite element method (FEM) to
understand the complete field response of complicated composites [18, 19]. But it still
lacks a robust mathematical foundation for the direct homogenization method. The
asymptotic homogenization (AH) can effectively extract the properties of composites
with heterogeneous components based on the small parameter perturbation theory, in
which a characteristic field similar to the displacement is presented [20]. This method
quickly became one of the most popular approaches in property prediction for lattices
[21] and periodic materials [22]. However, its low computational efficiency makes it
challenging to design microstructure with enormous numbers of the element [23]. In
3
addition, the compulsory periodic boundary conditions on the opposite surface in finite
element analysis make numerical simulation and optimization complex [24].
Moulinec and Suquet [25, 26] firstly developed an alternative computational
homogenization method based on Fast Fourier Transform (FFT) in the 1990s. However,
the convergence of the iterative scheme using the Green operator for the reference
medium in this method becomes extremely slow when the property contrast of
components is high [27]. Using accelerated iterative schemes of the Lippmann–
Schwinger equation can avoid such insufficiency [27-29]. Willot et al. [30]
modified the Green operator involved in FFT-based homogenization scheme by
centered differences method to achieve superior convergence rate. Some other
equivalent methods, like the Galerkin method [31] and a displacement-based approach
[32], also improve the efficiency of the FFT-based homogenization method. Zeman et
al. [33] established an FFT-based numerical method, starting from the weak form,
proceeding to the Galerkin discretization and the numerical quadrature, to solve general
history-dependent and time-dependent nonlinear material models. Vondřejc et al. [34]
recently developed a low-rank approximation-based solver for the Fourier–Galerkin
method [35] to accelerate the homogenization computing of complex heterogeneous
materials.
Compared with the AH method, the computational time complexity of the FFT-
based method for a problem with large N degrees of freedom significantly reduces from
O(N2) to O (NlogN) [36]. More importantly, this method essentially contains the
periodic boundary of the representative volume element (RVE) [37]. In addition, it
requires less memory allocation [3] and can use images/micrographs straightforward as
input models [27]. The FFT-based homogenization approach and its variants have built
an impeccable reputation in property prediction. Nowadays, they are widely used in
studying elastic–viscoelastic behaviors of polycrystalline materials [38], crack
propagation of composite laminates [39], electrical properties of highly-contrasted
composites [40, 41], and mechanical performances of 3D printing lattice [42]. To et al.
[43] developed the FFT-based numerical homogenization method to obtain effective
conductive properties of porous materials. Chen et al. [3, 44] also have applied FFT-
based homogenization to investigate the effective mechanical properties of novel
cellular structures with the triply periodic minimal surface and negative Poisson’s ratio.
After understanding the dependence of material properties on its microstructure,
researchers were keen to design novel materials by devising their material distribution.
Topology optimization strives to achieve the optimal distribution of material within a
design domain while maximizing specific mechanical properties under specified
constraints [45]. Its combination with the AH method incubates the inverse
homogenization method [46], which was proposed by Sigmund and Torquato [47] in
the 1990s to design uncommon materials with negative Poisson’s ratio and negative
thermal expansion. Diaz et al. [48] used the topology optimization method based on
solid isotropic material with penalization (SIMP) model to design metamaterials with
negative permeability. Zhou et al. [49] developed a systematic method to design the
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multi-phase microstructural composites with tailored thermal conductivity on Milton–
Kohn bounds. Huang et al. [50] have applied the bi-directional evolutionary structural
optimization (BESO) method and AH approach to design the microstructure of cellular
materials for maximum bulk and shear modulus. Zhang et al. [24] proposed the
alternative strain energy method to get the same accuracy as the AH method and used
it to design the microstructure with specific properties. Cadman et al. [51] critically
reviewed the optimal design of periodic microstructural composite materials with
specific properties like elastic stiffness, Poisson’s ratio, thermal expansion coefficient,
thermal conductivity, fluidic permeability, particle diffusivity, electrical conductivity
permittivity, and magnetic permeability. Wang et al. [52] proposed a new topological
shape optimization method for systematic computational design of a type of mechanical
metamaterials with negative Poisson’s ratios, which integrates the numerical
homogenization approach into a powerful parametric level-set topology optimization
method. Sivapuram et al. [53] linearized objective and constraint functions at every
iteration by sensitivity from AH and used the integer linear programming to address
binary microstructural optimization, achieving various non-volume microstructural
constraints with discrete (0/1) design variables. Collet et al. [54] introduced stress
responses within a density-based topology optimization framework applied to the
design of periodic microstructures. Xu et al. [55] combined the isogeometric topology
optimization method and the AH method to effectively design the ultra-lightweight
architected materials. Zheng et al. [56] recently proposed a new robust algorithm based
on the BESO method to get the auxetic microstructures with negative Poisson’s ratio.
Dos Reis et al. [57] combined AH scheme with genetic algorithms to elaborate the
inverse engineering of metamaterials contained in the target compliance tensor.
However, no one has attempted to take advantage of the outstanding virtues of FFT-
based homogenization to design advanced materials. This work develops a BESO
method to explore the FFT-based inverse homogenization for topology optimization of
microstructures. The elemental sensitivity of constitutive constants is established by the
strain energy under prescribed strain to determine the evolutional state of every element.
The rest of the paper is organized as follows. In Section 2, the FFT-based
homogenization theory and strain energy method are introduced. Section 3 exhibits the
implementation of the microstructural optimization process using the BESO method.
Section 4 gives several numerical examples and discussions in 2D and 3D. Conclusions
are drawn in Section 5.
2. Problem statement
The effective properties of architected materials depend on not only the constituent
materials but also their microstructure. If the materials are constructed by the
periodically unit cells or RVEs occupied in a Lebesgue open set Ω∈Rd (d = 2, 3 in 2D
and 3D, respectively), shown in Fig. 1. Maximizing the effective bulk or shear modulus
for a material can be implemented as an optimization problem defined as:
5
H
*
10,
Max F f
subject to d V
C
x x (1)
where ρ(x) is the relative density at point x. V* represents the volume fraction target of
solid material, which usually is set as an equality constraint V - V* = 0, where 0 < V*<
1 is a positive constant. The design variable ρ of this optimization problem is the volume
fraction of local material. It is ρ = 1 for a solid element while ρ = 0 for a void element.
To avoid the singularity of finite element analysis in optimization, a small value ρ =
ρmin is used to substitute with ρ = 0.
Fig. 1. Microstructure of a typical periodic cellular material and its RVE. The boundary 𝜕Ω of the
unit cell satisfies the periodic boundary condition. Domain Ω contains solid phase Ω1 and void phase
Ω2.
In an RVE, the effective elasticity tensor of periodic cellular material is defined as
-1
H1 1
,
d d
C σ x x ε x x (2)
where the local Cauchy stress tensor σ is calculated by the local strain ε in accordance
with Hooke’s law, given as:
.
σ C x : ε (3)
The local elasticity tensor could be C(ρ) = C0 or C(ρ) = 0, representing that element
with a centroid at an arbitrary location x is occupied by a homogeneous solid material
with ρ = 1, or a void material with ρ = 0, respectively. For the sake of simplicity, the
dependence on ρ and x is not explicitly expressed in the following formula of this
section.
The problem to solve Eq. (2) consists in finding the local strain and stress fields in
the domain Ω, which fulfill compatibility and linear momentum balance, respectively,
under periodic boundary conditions in such a way that the volume average of the strain
field equals the prescribed macroscopic strain. The governing equations of this problem
are
6
0
0
,
,
with
σ x
ε x ε
ε x # x
σ x n x - # x
(4)
where ▽ is the divergence operator, <·> denotes the spatial mean over Ω, # represents
periodicity, n(x) is the local unit normal vector at the boundary ∂Ω of the periodic
domain Ω, and -# denotes anti-periodicity. The external loads are then introduced as an
average macroscopic strain ε0, and the local strain field composes of two items as
0
,
ε = ε
ε
(5)
where
ε
is the periodic fluctuation strain field and satisfies
0
ε
.
By introducing the homogeneous reference material with elastic tensor C0, Eq. (3)
can be rewritten as
0 0 0
,
σ = C : ε = C - C : ε + C : ε τ C :
ε
(6)
where the polarization stress tensor τ = (C-C0): ε. Introducing the Cauchy stress Eq. (6)
and strain field Eq. (5) into the equilibrium equation in Eq. (4), the following
homogeneous partial differential equation (PDE) is obtained
0
.
C : ε =
τ
(7)
When the periodic fluctuating strain field is expressed derivative in terms of a periodic
fluctuating displacement field
u
(
ε
u
) as a compatible form, and introduced into
Eq. (7), we get
0
.
C : u =
τ
(8)
If we can obtain the basic Green’s function G0 of the resulting equation, Eq. (8) can be
solved by using the Green’s function method, and the fluctuating displacement under
the dummy force ▽τ can be derived as
0
x ( ') ' ',
d
u G x x τ
x x
(9)
where Green’s function G0(x-x') represents the displacement vector at point x under the
unit force at point x'.
Therefore, taking the symmetric gradient of Eq. (9) and applying the divergence
theorem and the corresponding product symmetries, the fluctuating strain field is
directly obtained as a function of the polarization field by
0 0
,
( ') ' '= ( ') ' ',
ij kl ijkl
G d d
ε = x x τ x x x x τ
x x
(10)
where the fourth-order tensor operator Γ0 is defined as
0 0
,
( ')= ( ').
ijkl ij kl
G
x x x x
(11)
7
Therefore, the local strain field can be expressed by Lippmann-Schwinger equation as
0 0
= ,
ε ε Γ τ
ε
(12)
where ‘*’ represents convolution operation. Lippmann-Schwinger equation Eq. (12)
can be rather easily solved in Fourier frequency space using the property of convolution,
and then the problem is strongly simplified leading to the expression
01 0 0 :
ˆ
: ,
F F
ε ε Γ C C
ε
(13)
where the Fourier transform operator and its inverse operator are represented as F and
F-1, respectively. The macroscopic strain ε0 is prescribed into each node of the unit cell
in local strain analysis. The details of Eq. (13) can be found in the Appendix.
To illustrate the algorithm more clearly, the formulas in the following text are in
Voigt notation, and the computations are described in the form of matrices and vectors.
For a 2D isotropic composite, three linearly independent unit strains ε01 = [1, 0, 0]T, ε02
= [0, 1, 0] T, and ε03 = [0, 0, 1]T are applied to retrieve all members of effective stiffness
matrix. As the unknown strain field at the left and right sides of Eq. (13), the fixed-
point method was first proposed to solve the equation [25]. This paper solves the linear
algebra problem derived from Eq. (13) by the conjugate gradient method [27]. After the
local strain field has been obtained, the stresses in every point/element can be calculated
using Eq. (3). Therefore, the local strain and stress field of a periodic unit cell under
prescribed strain can be obtained by the FFT-based homogenization method.
As periodic boundary conditions have been naturally contained in the FFT-based
homogenization approach, the effective constitutive matrix can be obtained by Eq. (2).
However, specific moduli cannot be separately expressed. The strain energy-based
approach has been adopted to extract the objective moduli and used to conduct the
sensitivity analysis of composites [58]. The method also has been applied in
microstructure topology optimization to avoid the complexity of the asymptotic
homogenization method [59].
If considering the strain energy induced by the test strain ε03, we get an objective
function equivalent to the shear modulus G
H H
1111 1122 1112
T
0 H 0 H H H H
3 3 1212 2211 2222 2212
H H H
1211 1222 1212
0
0,0,1 0 .
1
H
C C C
F G C C C C
C C C
ε C ε
(14)
With the same idea, the objective functional equivalent to the bulk modulus K is
obtained by using the test strain εt1 = [1, 1, 0]T, given as:
H H H H TH
1111 2222 1122 2211
1 1 1
1 1
= ,
4 4 2
t t t
C C C C
F K E
ε C ε
(15)
where
1
t
E
denotes the total strain energy of the unit cell under the test strain. For an
isotropic material, the bulk modulus in 3D can be expressed by
8
3T
H H
2 2 2
, 1
1 1 2
= ,
9 9 9
iijj t t t
i j
F K C E
ε C ε
(16)
where the test strain is εt2 = [1, 1, 1, 0, 0, 0]T. The average shear modulus under three
directions is determined by
H H H 5TH
1212 2323 3131
3
1
,
3 3
ti ti
i
C C C
F G
ε C
ε
(17)
where the test strains are εt3 = [0, 0, 0, 1, 0, 0]T, εt4 = [0, 0, 0, 0, 1, 0]T and εt5 = [0, 0, 0,
0, 0, 1]T.
Therefore, using the strain energy-based method, the relation between total strain
energies under specific prescribed strain and constants in elastic tensor can be
constructed to simplify the expression of objective moduli, especially for the bulk
modulus. The complex objective functions like Poisson’s ratio or specific elastic tensor
can be transformed into combinations of total strain energies under specific test strains.
The total strain energy of unit cell under a test strain can be computed by
T T
1
1 1
= ,
2 2
n
t i i
i
E d
ε C ε x ε
σ
(18)
where ρi is the density of element-i, n is the total number of elements, and the local
strain εi and the local stress σi can be extracted from the FFT-based homogenization
computation.
3. Numerical implementation
In the SIMP topology optimization approach, the local constitutive tensor can be
related to the density variable in the power law. The density means how much solid
material should be endowed to the current element. The penalization material model
has been utilized by the majority of topology optimization methods. The simple
penalization material function is
0
,
p
i i i
C C
(19)
where C0 is the elastic tensor of solid material. In this paper, the BESO technique is
adopted to approach the topology optimization of microstructure with extreme
mechanical properties. In the BESO method, the design variables are discrete values
which are either ρ = ρmin for a void element or ρ = 1 for a solid element.
As the objective function is a combination of total strain energies in Eq. (18) under
specific test strains, the solution for the sensitivity of the objective function can be
converted into the sensitivity of strain energy. The adjoint method is adopted to
determine the sensitivity of the total strain energy in Eq. (18), as explained below. By
introducing a vector of Lagrangian multiplier λ, an extra term λ(Cε - σ) can be added to
the total strain energy without changing anything due to Eq. (3). Thus, the new strain
energy is
9
T
1
.
2
E
ε σ Cε
σ
(20)
The sensitivity of the modified strain energy can be written as
T
T
1 1
+ + + .
2 2
i i i i i i i
E
ε σ C ε
σ
σ ε Cε σ ε C
(21)
It is noted that the third term in Eq. (21) becomes zero due to the constitutive equation
Eq. (3). Also, it is assumed that the variation of an element has no effect on the applied
test strain vector in the FFT-based homogenization method and therefore
0
i
ε.
Thus, the sensitivity of the total strain energy becomes
T
1
+ .
2
i i i
E
σ
C
ε
ε
(22)
In Eq. (20), the Lagrangian multiplier vector λ can be freely chosen as Cε-σ is equal
to zero. To eliminate the unknown
i
σof the sensitivity expression in Eq. (22), λ
is chosen such that
T
1
=0.
2
ε (23)
By substituting λ = 1/2εT into Eq. (22), the sensitivity of the objective function
becomes
T
1
.
2
i i
E
C
ε
ε
(24)
Based on the assumption that the change in local density only affects the local stiffness,
the sensitivity of the objective function si with regard to the change in the element-i for
the total strain energy can be expressed as
T
1
2
i
i i i
i i
E
s
C
ε
ε
(25)
By substituting the material interpolation scheme Eq. (19) into the above equation, the
sensitivity can be found as
T 1 T
0 0
1 1
,
2 2
p p
i i i i i i i i
i i
p p
s p E
ε C ε ε C ε (26)
where T
0
1
2
p
i i i i
E
ε C
ε
is the local strain energy of element-i. It turns out that the
sensitivity of the total strain energy for an RVE with respect to the element density
variable is equal to the product of the scaling factor and the strain energy of the
concerned element. The final expression of strain energy sensitivity in this paper is the
same as the formula derived by the FEM-based scheme [24]. The total energy
10
sensitivity related to the elemental variable can be transformed to be the elemental
sensitivity. And, local strain and stress of the element can be obtained from the FFT-
based homogenization method to compute the elemental strain energy.
To solve the checkerboard patterns and mesh-dependency problems when the unit
cell is discretized into a large number of elements, the mesh-independency filter from
the digital image processing field is applied, which can average the elemental sensitivity
numbers with its neighboring elements [60]. Therefore, the elemental sensitivity
number will be modified by the following equation:
1
1
ˆ
,
N
ij j
j
iN
ij
j
w r s
s
w r
(27)
where sj is original sensitivity, 𝑠 is filtered sensitivity, rij is the distance between the
center of the element i and j, and w denotes the weight factor. Weight factor w can be
determined by
min min
min
,
0,
ij ij
ij
ij
r r r r
w r r r
(28)
where rmin is the filter radius. This procedure is utilized to filter the sensitivity
information.
The historical information can be used to get the recursion of sensitivity number,
making the iterative procedure more stable as a common approach in the BESO method.
Therefore, the current sensitivity number of the kth iteration is given by
1
1
ˆˆˆ
.
2
k k k
i i i
s s s
(29)
When the current volume fraction does not reach the target V*, the next volume
fraction (k+1th iteration) is determined by the current volume fraction as follows
1
1 ,
k k
V V ER
(30)
where ER denotes the evolutionary ratio, and V0 is the volume fraction of the initial
design.
When the next volume fraction is determined, the dichotomy method is applied to
find the threshold of sensitivity
th
ˆ
k
s
for approaching the density variable in every
element of the next design. When the elemental sensitivity is lower than the threshold
value:
th
ˆ
ˆ
k k
i
s s
, the element will be set void. Otherwise, the element will be set solid.
The convergence criterion is according to the tolerance of objective modulus as
11
1 1
1
1
1
,
Nk i k N i
i
Nk i
i
M M
M
(31)
where M is the objective modulus, k is the current iteration, N is recall numbers, and ζ
is allowable convergence error. N and ζ are set to be 5 and 10-3, respectively, throughout
this paper. This implies that the change of the objective function over the last 10
iterations is acceptably small (10-3).
The BESO technique is mathematically simple and easy to implement. And the
local strain and stress values are the only input data for the BESO technique to conduct
the topology optimization [50]. There are no intermediate densities in BESO, which
contributes to generating binary structures. Therefore, the whole flowchart of
microstructure topology optimization is exhibited in Fig. 2, combined with the FFT-
based homogenization method and BESO technique.
Fig. 2. The flowchart of the FFT-based inverse homogenization for cellular material Design.
Computing
ε
and
σ
by
FFT-based homogenization
Compute the elemental
sensitivity s
ik
by strain
energy-based method
BESO procedure
The k
th
topology
No
Yes
0
Determine material of
every element
th
ˆ ˆ
, 1
k k
i i min i
if s s else
；
Filter and stabilize
elemental sensitivity,
get
ˆ
k
i
s
The volume fraction
Convergence ?
The threshold of
sensitivity
Is volume constraint
satisfied?
Start
Initial topology
End
Optimal topology
No
Yes
th
ˆ
k
s
12
4. Examples and discussion
In this section, we firstly conducted some examples in 2D and 3D for maximizing
moduli. Young’s modulus and Poisson’s ratio of solid material are set as 1 and 0.3,
respectively. In Sections 4.1 and 4.2, the initial design of 2D or 3D topology
optimization examples except for the scaffold is that void material is assigned to the
center element of square or cubic, and solid material is given the rest elements. In these
cases, the moduli of optimal microstructure are frequently compared with known upper
bounds of bulk and shear moduli. The upper and lower bounds for isotropic composites
are derived by Hashin and Shtrikman (HS) [61]. For cellular material with volume
fraction or relative densityρ, the upper HS bounds of bulk modulus (KHSU) and shear
modulus (GHSU) in 2D plane stress case are defined as:
s s
HSU
s s 1
G K
KG K
(32-a)
s s
HSU
s s s s
2 2 2
G K
GG K K G
(32-b)
and in the 3D case are defined as:
s s
HSU
s s
4
4 3 1
G K
KG K
(33-a)
s s s
HSU
s s s s
9 8
20 15 6 2
K G G
GG K K G
(33-b)
where Ks and Gs represent the bulk and shear moduli of solid phase, respectively.
4.1. 2D numerical examples
For these 2D examples, the square design domain of the unit cell is discretized into
201×201 elements. Evolutional ratio ER is 0.02, filter radius rmin is 10, pseudo density
ρmin is 0.01, and penalty exponent p is 3. Bulk modulus and shear modulus of the 2D
solid phase are 0.714 and 0.385, respectively.
4.1.1. Maximization of bulk modulus
The topology optimizations of 2D microstructure for maximizing bulk modulus are
performed under the volume fraction constraints of 20%, 30%, 40%, and 50%,
respectively. The numbers of these iterations are 90, 71, 50, and 45, respectively. And
the corresponding optimal bulk moduli are 0.055, 0.088, 0.125, and 0.180, respectively.
As shown in Fig. 3(a-d), the optimal microstructures exhibit the square pattern with
different details. The optimal values of the objective function in this paper approach the
results of Huang et al. [50], but their optimal microstructures are different due to the
non-unique solution of topology optimization. Fig. 3(f) shows the evolutional histories
of bulk modulus and volume fraction with respect to iteration when the volume fraction
13
constraint is 30%. The results of 40% and 50% volume fractions constraints can be
found in the iterative histories, meaning the evolutional procedure converges into
optimal results at every step. The optimal results are compared with the HS upper
bounds, as illustrated in Fig. 3(g). Bulk moduli of optimal microstructures agree well
with the HS upper bound.
Fig. 3. The RVEs of 2D cellular materials with maximal bulk modulus under volume fraction of (a)
20%, (b) 30%, (c) 40%, (d) 50%. (e) RVEs of 30% volume fraction ranked in a 2×2 array. (f) The
convergence history of an example under 30% volume fraction constraint. RVE is plotted in
iteration history. (g) The effective bulk modulus and the HS upper bound.
4.1.2. Maximization of shear modulus
Similarly, topology optimizations of microstructure for maximizing the shear
modulus are conducted under the volume fraction constraint of 20%, 30%, 40%, and
50%, respectively. Their corresponding numbers of iterations are 85, 67, 50, and 44,
respectively. Maximum bulk moduli are 0.051, 0.078, 0.106, and 0.139, respectively.
The optimal microstructures show the diamond pattern as shown in Fig. 4(a-c). A small
interspace appears in the rhombic microstructure when the volume fraction constraint
is 50%, as exhibited in Fig. 4(d). When we double the filter radius in the topology
optimization parameter settings, these small features disappear, and the optimal RVE
also shows a rhombic structure.
14
The optimal microstructures of this paper for maximizing shear modulus are similar
to the results from Xia et al. [59]. The optimal results are compared with the HS upper
bounds depicted in Fig. 4(f). Shear moduli of the optimal microstructures are slightly
above the HS upper bounds, which is attributed to the non-isotropy of the
microstructure.
Fig. 4. The RVEs of 2D cellular materials with maximal shear modulus under volume fraction of (a)
20%, (b) 30%, (c) 40%, (d) 50%. (e) RVEs of 30% volume fraction ranked in a 2×2 array. (f) The
effective shear modulus G and the HS upper bound.
4.1.3. Negative Poisson’s ratio
The design of architected materials with negative Poisson’s ratio v12 = C1122/C1111
or v21 = C2211/C2222 using topology optimization is still a challenging subject [62, 63].
In this paper, using the direct definition of Poisson’s ratio, the maximum object F = -
(C1122+ C2211)/(C1111+ C2222) is constructed to achieve negative Poisson’s ratio property
considering both values of Poisson’s ratio. Then, the partial derivative of the objective
function with respect to density is:
H H H H
2
1111 2222 1122 22 11
1122 22 11 1111 2222 1111 2222
C C C C
FC C C C C C
(34)
and the sensitivity of object can be simplified as
S
h
e
a
r
m
o
d
u
l
u
s
,
G
(a) (b) (c) (d)
(e)
(f)
15
H H H H
1111 222 2 1122 2 211
1122 2 211 1 111 2222
.
C C C C
s C C C C
(35)
Specific strain energy can be used to replace the homogenized elastic constants in Eq.
(35) as follows
H H H H
1111 2222 1 6 1122 2 211 1 6
, ,
t t t t
C C E E C C E E
(36)
where
6
t
E
is the strain energy of unit cell under εt6 = [1, -1, 0]T.
Then, the FFT-based inverse homogenization method is employed to conduct the
optimization with objective function F = - (C1122 + C2211)/(C1111 + C2222) to obtain the
auxetic microstructure. The same optimized settings with the above sections are
adopted, and the results under 50% volume fraction are shown in Fig. 5. The final
optimal microstructure is depicted in Fig. 5(a). In its 4×4 structure, we also can find its
equivalent RVE, which is a rotating microstructure with negative Poisson’s ratio
property, as exhibited in Fig. 5(b). The method gives a rotating negative Poisson’s ratio
materials rather than a re-entrant structure generated from topology optimization as
Zheng et al. [56]. And the calculated Poisson’s ratio values of the optimized unit cell
are v12 = -0.4264 and v21 = -0.4238, with good symmetry. This example shows that the
FFT-based inverse homogenization method can generate well-designed negative
Poisson's ratio materials without defining additional constraints.
Fig. 5. The 2D cellular materials with negative Poisson’s ratio: (a) RVE under volume fraction of
50%; (b) RVEs ranked in a 4×4 array; (c) elasticity matrix. A red dashed square marks the rotating
RVE.
4.2. 3D numerical examples
For the general 3D numerical examples, the design domain of the unit cell in shape
with cubic is discretized into 51×51×51 elements. Evolutional ratio ER is 0.04, filter
0.1010 -0.0431 0.0000
-0.0431 0.1017 0.0000
0.0000 0.0000 0.0235
(a)
(c)
(b)
16
radius rmin is 5, pseudo density ρmin is 0.01, and penalty exponent p is 3. Bulk modulus
and shear modulus of the 3D solid phase are 0.833 and 0.385, respectively.
4.2.1. Maximization of bulk modulus
The topology optimizations of 3D microstructure for maximizing the bulk modulus
are performed under the volume fraction constraints of 10%, 20%, 30%, and 40%,
respectively. The numbers of these iterations are 78, 85, 44, and 66, respectively. And
the corresponding optimal bulk moduli are 0.0152, 0.0436, 0.1084, and 0.1581,
respectively. As demonstrated in Fig. 6(b-d), the optimal microstructures exhibit the
cubic foam pattern. In contrast, RVE under a volume fraction constraint of 10% gives
a cubic lattice shown in Fig. 6(a). Fig. 6(f) reports the evolutional histories of bulk
modulus and volume fraction concerning iteration when the volume fraction constraint
is 30%. Both bulk modulus and volume fraction are stably convergent. As shown in Fig.
6(g), bulk moduli of optimal microstructures are close to the HS upper bounds with a
slight deviation. Bulk moduli of optimal microstructure under 10% and 20% volume
fractions are slightly below the HS upper bounds. Due to the change in topological
mode, the optimal structure with volume fraction constraints of 30% or 40%, presenting
a closed cubic [44, 61], can achieve higher values of bulk modulus than HS upper
bounds.
Fig. 6. The RVEs of 3D cellular materials with maximal bulk modulus under volume fraction of (a)
10%, (b) 20%, (c) 30%, (d) 40%. (e) RVEs of 30% volume fraction ranked in a 2×2×2 array. (f) The
convergence history of an example under 30% volume fraction constraint. (g) The effective bulk
17
modulus and the HS upper bound.
4.2.2. Maximization of shear modulus
The topology optimizations of microstructure for maximizing shear modulus are
conducted under the volume fraction constraint of 10%, 20%, 30%, and 40%,
respectively. The numbers of these iterations are 184, 82, 73, and 62, respectively. And
the corresponding optimal bulk moduli are 0.0097, 0.0368, 0.0618, and 0.0897,
respectively. As shown in Fig. 7(b-c), the optimal microstructures are similar to the
microstructure based on Schwarz primitive surface whose superior shear modulus has
been demonstrated by Chen et al. [44]. At a volume fraction of 40%, the microstructure
tends to evolve into a foam in shape with a cuboctahedron (comprised of six squares
and eight equilateral triangles), as illustrated in Fig.7(d). In comparison, the optimal
microstructure degrades to be lattice structure when the volume fraction constraint is
10%, as exhibited in Fig. 7(a). The optimal shear modulus values and the corresponding
HS upper bounds are shown in Fig. 7(f). When the volume fraction constraint is large,
shear moduli of optimal microstructures approach the HS upper bounds with a slight
difference. When the constraint of volume fraction is 10%, the microstructure
degenerates into lattice structure resulting in weakness of shear modulus.
Fig. 7. The RVEs of 3D cellular materials with maximal shear modulus under volume fraction of
(a) 10%, (b) 20%, (c) 30%, (d) 40%. (e) RVEs of 30% volume fraction ranked in a 2×2×2 array. (f)
The effective shear modulus and the HS upper bound.
4.2.3. Maximization of comprehensive modulus
The isotropic cellular structure with ultimate stiffness tends to approach the HS
upper bound on various moduli [44]. Since the above 3D examples only consider the
18
optimization of a single modulus, the weighted moduli sum method is introduced in
this section to optimize the microstructure with high stiffness and isotropy. The
comprehensive modulus coefficient Ec = K/KHSU + G/GHSU is defined as the objective
function to balance the difference between shear and bulk modulus.
The topology optimizations of microstructure for maximizing the comprehensive
modulus are conducted under the volume fraction constraint of 10%, 20%, 30%, and
40%, respectively. At a volume fraction of 10%, the optimal microstructure is an
elaborate lattice, as shown in Fig. 8(a-b). Under volume fraction of 20%, 30%, and 40%,
we found all optimal results are similarly closed foam microstructures as exhibited in
Fig. 8(c-e). From the inside view of the unit cell, the microstructure looks like a
polyhedral pore structure. From the outside, it shows a cubic+tetradecahedral structure.
The microstructure pattern reflects the advantages of the composite as the cubic
structure has maximum bulk modulus, and the tetradecahedral structure has maximum
shear modulus concluded from the previous results. Taking the 30% volume fraction
constraint as an example shown in Fig. 8(f), the comprehensive modulus coefficient
gradually approaches convergence as the volume fraction stabilizes in the iterative
process.
Fig. 8. The 3D cellular materials with maximal comprehensive modulus under volume fraction of
10%: (a) the RVE and (b) its half-section. The optimization results at 30% volume fraction: (c) the
RVE and (b) its half-section; (e) RVEs ranked in a 2×2×2 array; (f) the convergence history.
Zener ratio “Z” is generally adopted to characterize anisotropy index of architected
materials [61]. Conceptually, the indicator quantifies how far a material is from being
isotropic and is mathematically expressed as:
19
H H
1111 1122
1212
.
2 1 2 H
C CE
ZCG v
(37)
Obviously, Z = 1 means isotropic material because of E = 2G(1+ν). In this section, the
changes of Zener ratio during the optimization process of 20% and 30% volume fraction
are shown in Fig. 9. The values of Zener ratio are always around ~1 during the
optimization process. In Fig. 9, the 3D directional dependence of Young’s modulus for
the final microstructure exhibits a spherical shape, meaning it is ideal isotropic.
Under the constraints of 20%, 30%, and 40% volume fraction, the final Zener ratios
of the optimal microstructures are 0.996, 0.927, and 0.902, respectively. It can be seen
that the weighted moduli sum approach proposed in this section can obtain the
approximately isotropic microstructure, which avoids the complicated expression of
isotropic constraint in previous research [64]. The comparisons of shear modulus and
bulk modulus of the optimal structure with the HS upper bound are depicted in Fig. 10.
Both shear and bulk modulus vary according to the HS upper bound, and bulk modulus
is closer to the HS upper bound. Under the volume fraction constraints of 20%, 30%,
and 40%, the final comprehensive modulus coefficients Ec of the optimized unit cell
are 1.601, 1.717, and 1.811. All average values of the ratio of both moduli to the HS
upper bound exceed 0.8. The numerical examples show that the FFT-based inverse
homogenization approach with comprehensive modulus can achieve a microstructure
of high moduli with the approximately isotropic property.
Fig. 9. Zener ratio in iterative histories under volume fraction constraint of: (a) 20% and (b) 30%.
Young’s modulus surface of the optimal RVE are plotted at final iteration.
20
Fig. 10. Comparison between the results of this paper and HS upper bound for: (a) bulk modulus
and (b) shear modulus.
In order to use finer voxels to depict the microstructure, the unit cell is discretized
into 101×101×101 elements. Other optimization parameters keep the same as the
previous 3D example, and then the numerical example under the volume fraction
constraint of 30% is carried out. As exhibited in Fig. 11, the microstructure shows multi-
cavities in shape with cubic, octahedron, tetrakaidecahedron, and sphere. Compared to
the optimal microstructure with 51×51×51 elements, many more structural details are
captured in Fig. 11. The effective elasticity matrix of the optimal microstructure is
displayed in Fig. 11(d), and the corresponding Zener ratio is 0.924. In Fig. 11(d),
Young’s modulus surface of the final optimal microstructure shows a spherical shape,
which means the structure is ideal isotropic. Moreover, Ec of the optimal microstructure
is 1.834, which exceeds the values of the low-resolution example. Because the optimal
microstructure is at high resolution, the manufacturable geometry can be generated by
simply smoothing, as exhibited in Fig. 11(e).
0 0.1 0.3 0.5
Volume fraction, V
f
0
0.05
0.1
HS upper Bound
Results of this paper
0 0.1 0.3 0.5
Volume fraction, V
f
0
0.05
0.1
0.15
0.2
HS upper Bound
Results of this paper
(a) (b)
21
Fig. 11. The 3D cellular materials with fine voxel for maximizing comprehensive modulus under
30% volume fraction constraint: (a) the RVE and (b) its half-section. (c) RVEs ranked in a 2×2×2
array. (d) Elasticity matrix and Young’s modulus surface. (e) Geometry of RVE after smoothing.
4.2.4. Scaffold design
In this section, we consider using the FFT-based inverse homogenization method
to tailor the entries of the elastic tensor, aiming to attain those of a human bone [65].
Such an optimization problem will yield a scaffold to provide a temporary, artificial
extracellular matrix for neo-tissue growth, which should ideally resemble the
mechanical properties of the host bone [65]. Therefore, an objective function is often
formulated in terms of the differences between the corresponding terms of the targeted
stiffness C* and homogenized values CH, subjected to a mating porosity or relative
density (V*) to the host tissue as:
32
* H
, 1
*
10.
ijij ijij
i j
Min F C C
subject to d V
x x
(42)
The sensitivity of the objective function with respect to the design variable can be
derived as follows:
H
3
H *
, 1
2 .
ijij
ijij ijij
i j
C
F
s C C
(43)
22
The base material used in the scaffold design is isotropic with Poisson’s ratio of 0.3
and Young’s modulus of 6 GPa. The targeted elastic constants in the elasticity matrix
for the scaffold can be found in Table 1. The FFT-based inverse homogenization method
is employed to conduct the optimization design of the scaffold. The thick cubic lattice
is adopted as the initial microstructure of topology optimization [65], shown in Fig.
12(a). The optimization scaffold under 50% volume fraction is exhibited in Fig. 12(b-
d). This design provides a spatially continuous and open structure that is conducive to
bone regeneration. The non-complete symmetry of microstructure is due to the
anisotropy of the mechanical target. As the optimization proceeds, the optimization
objective quickly converges to zero and keeps stable, as illustrated in Fig. 12(e).
The comparisons between the targeted elastic constants and optimization results are
listed in Table 1. The largest deviation between two sets of data is 1.38%. There is good
agreement between the elastic constants of the optimized result and the target. The
constants in the elasticity tensor of the final scaffold are only slightly lower than that of
the target.
Fig. 12. Scaffold optimization. (a) The initial design, (b) the optimal RVE, and (c) its half-section.
(d) RVEs ranked in a 2×2×2 array. (e) Convergence history.
Table 1
The entries of an effective elastic tensor and their targets for the optimization of a scaffold.
Elastic entry Target (GPa) Effective value (GPa) Deviation (%)
C1111 2.05 2.0454 0.22
C2222 2.025 2.0110 0.69
C3333 1.05 1.0484 0.15
C1212 0.75 0.7455 0.60
23
C1313 0.5 0.4942 1.16
C2323 0.5 0.4831 1.38
5. Conclusions
This paper proposes a new inverse homogenization method for designing cellular
materials with extreme mechanical properties or specified elastic tensors. The FFT-
based homogenization, which naturally satisfies periodic boundary conditions, can
retrieve the effective constitutive properties of an RVE with up to one million elements.
Also, we simplify the optimization objective as strain energy under specific test strain,
making the numerical implementation of BESO straightforward and efficient.
With a single constraint on volume fraction, this approach allows maximizing or
minimizing objective functions constituted by homogenized elasticity constants such
as bulk modulus, shear modulus, and Poisson’s ratio. We found that a series of 2D and
3D cellular structures like Schwarz Primitive materials can attain HS bound of bulk and
shear modulus.
Also, cellular structures overlapped by a cuboid and a tetradecahedron but
imbedded by an internal void polyhedron exhibit maximal comprehensive modulus.
Such a topology is independent of the volume fraction and nearly attains isotropy.
Besides, the proposed method tailors the elastic tensors to those of a human bone,
supporting cell attachment and tissue growth.
The approach presented in this work facilitates the design of architected cellular
material with extreme mechanical performance and widens the application range of
FFT-based homogenization.
Declaration of competing interest
The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported in this
paper.
CRediT authorship contribution statement
Zeyao Chen: Methodology, Software, Validation, Writing – original draft.
Baisheng Wu: Conceptualization, Supervision, Writing – reviewing and editing. Yi
Min Xie: Investigation, Writing – reviewing and editing. Xian Wu: Supervision,
Investigation. Shiwei Zhou: Conceptualization, Writing – reviewing and editing.
Acknowledgments
The authors thank the anonymous reviewers for their invaluable contribution in
assisting the authors to improve the paper. The authors at Guangdong University of
Technology would like to thank the support from National Natural Science Foundation
of China (No. 11672118) and Research and Development Plans in Key Areas of
24
Guangdong, China (No. 2019B090917002). The authors at RMIT University are
grateful for the support of the Australian Research Council (DP200102190). The
authors at Tongji University are grateful for the support provided by National Key R&D
Program of China (No. 2018YFB0105604).
Appendix
From Eq. (11), the projection tensor Гijkl in the real space can be derived by the
average of the double derivatives of Green’s function as
2 0 2 0
2 0 2 0
01
.
4
jl jk
il ik
ijkl
j k j l i k i l
G G
G G
x x x x
x x x x x x x x
(A-1)
In the frequency space, the projection tensor operator is expressed as
0 0 0 0 0
1ˆ ˆ ˆ ˆ
ˆ
,
4
ijkl il j k ik j l jl i k jk i l
G G G G
(A-2)
where ξ is the coordinate vector in the frequency domain.
The elastic tensor of the isotropic reference material with an average elasticity
tensor assumed as C0 can be written as
0 0 0
,
ijkl ij kl ik jl il jk
C
(A-3)
where Lamé parameters of reference material are represented as λ0 and μ0, and δ is the
Kronecker function. The Green’s function of elastic homogenous PDE in the frequency
domain can be obtained as
1
12
0 0 0 0 0
ˆ
( ) .
ij ijk l k l i j ij
G C
ξ
(A-4)
Therefore, Green’s function in the frequency domain can be written as
0 0
0
2 4
00 0 0
1
ˆ
( ) .
2
ij i j
ij
G
ξ ξ
(A-5)
Then, introducing Eq. (A-5) into Eq. (A-2), the Fourier expression of the periodic
Green’s projection operator Γ0 [25] in the frequency domain is given as
0 0
0
2 4
0 0 0
0
1
ˆ
2
4
.
i j k l
ijkl ik l j il k j jk l i lj i k
ξ ξ
(A-6)
The convergence rates of the FFT-based homogenization scheme depend on the choice
of the reference tensor C0. The optimal rate of convergence is achieved by defining the
reference material as follows:
0 0
min { }+ max{ } min { }+ max{ }
, .
2 2
(A-7)
References
[1] O. Al-Ketan, R. Rowshan, R.K. Abu Al-Rub, Topology-mechanical property relationship of 3D
25
printed strut, skeletal, and sheet based periodic metallic cellular materials, Addit Manuf, 19 (2018)
167-183.
[2] X. Zheng, H. Lee, T.H. Weisgraber, M. Shusteff, J. DeOtte, E.B. Duoss, J.D. Kuntz, M.M. Biener, Q.
Ge, J.A. Jackson, Ultralight, ultrastiff mechanical metamaterials, Science, 344 (2014) 1373-1377.
[3] Z. Chen, X. Wu, Y.M. Xie, Z. Wang, S. Zhou, Re-entrant auxetic lattices with enhanced stiffness: a
numerical study, Int J Mech Sci, 178 (2020) 105619.
[4] L. Valdevit, A. Pantano, H.A. Stone, A.G. Evans, Optimal active cooling performance of metallic
sandwich panels with prismatic cores, Int J Heat Mass Transf, 49 (2006) 3819-3830.
[5] S. Zhou, Q. Li, The relation of constant mean curvature surfaces to multiphase composites with
extremal thermal conductivity, J Phys D: Appl Phys, 40 (2007) 6083.
[6] A. Ajdari, H. Nayeb-Hashemi, A. Vaziri, Dynamic crushing and energy absorption of regular,
irregular and functionally graded cellular structures, Int J Solids Struct, 48 (2011) 506-516.
[7] Y.C. Zhou, L. Ye, W.Q. Chen, Impact mitigation performance of hybrid metamaterial with a low
frequency bandgap, Int J Mech Sci, 213 (2022).
[8] N. Sui, X. Yan, T.Y. Huang, J. Xu, F.G. Yuan, Y. Jing, A lightweight yet sound-proof honeycomb
acoustic metamaterial, Appl Phys Lett, 106 (2015).
[9] J.U. Surjadi, L. Gao, H. Du, X. Li, X. Xiong, N.X. Fang, Y. Lu, Mechanical metamaterials and their
engineering applications, Adv Eng Mater, 21 (2019) 1800864.
[10] G. Bertolino, M. Montemurro, Two-scale topology optimisation of cellular materials under mixed
boundary conditions, Int J Mech Sci, 216 (2022).
[11] T.A. Schaedler, W.B. Carter, Architected cellular materials, Annu Rev Mater Res, 46 (2016) 187-
210.
[12] M. Osanov, J.K. Guest, Topology optimization for architected materials design, Annu Rev Mater
Res, 46 (2016) 211-233.
[13] J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems,
Proc R Soc A, 241 (1957) 376-396.
[14] R. Hill, A self-consistent mechanics of composite materials, J Mech Phys Solids, 13 (1965) 213-
222.
[15] R.M. Christensen, K.H. Lo, Solutions for effective shear properties in three phase sphere and
cylinder models, J Mech Phys Solids, 27 (1979) 315-330.
[16] T. Mori, K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting
inclusions, Acta Metall, 21 (1973) 571-574.
[17] M.P. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a
homogenization method, Comput Methods Appl Mech Eng, 71 (1988) 197-224.
[18] T. Kanit, S. Forest, I. Galliet, V. Mounoury, D. Jeulin, Determination of the size of the representative
volume element for random composites: statistical and numerical approach, Int J Solids Struct, 40
(2003) 3647-3679.
[19] T. Nakamura, S. Suresh, Effects of thermal residual stresses and fiber packing on deformation of
metal-matrix composites, Acta Metall Mat, 41 (1993) 1665-1681.
[20] S. Ghosh, K. Lee, S. Moorthy, Two scale analysis of heterogeneous elastic-plastic materials with
asymptotic homogenization and Voronoi cell finite element model, Comput Methods Appl Mech
Eng, 132 (1996) 63-116.
26
[21] S. Arabnejad, D. Pasini, Mechanical properties of lattice materials via asymptotic homogenization
and comparison with alternative homogenization methods, Int J Mech Sci, 77 (2013) 249-262.
[22] D. Préve, A. Bacigalupo, M. Paggi, Variational-asymptotic homogenization of thermoelastic
periodic materials with thermal relaxation, Int J Mech Sci, 205 (2021) 106566.
[23] A.L. Kalamkarov, I.V. Andrianov, V.V. Danishevs’kyy, Asymptotic homogenization of composite
materials and structures, Appl Mech Rev, 62 (2009).
[24] W. Zhang, G. Dai, F. Wang, S. Sun, H. Bassir, Using strain energy-based prediction of effective
elastic properties in topology optimization of material microstructures, Acta Mech Sin, 23 (2007)
77-89.
[25] H. Moulinec, P. Suquet, A numerical method for computing the overall response of nonlinear
composites with complex microstructure, Comput Methods Appl Mech Eng, 157 (1998) 69-94.
[26] J.-C. Michel, H. Moulinec, P. Suquet, Effective properties of composite materials with periodic
microstructure: a computational approach, Comput Methods Appl Mech Eng, 172 (1999) 109-143.
[27] J. Zeman, J. Vondřejc, J. Novák, I. Marek, Accelerating a FFT-based solver for numerical
homogenization of periodic media by conjugate gradients, J Comput Phys, 229 (2010) 8065-8071.
[28] L. Gélébart, R. Mondon-Cancel, Non-linear extension of FFT-based methods accelerated by
conjugate gradients to evaluate the mechanical behavior of composite materials, Comput Mater Sci,
77 (2013) 430-439.
[29] J.C. Michel, H. Moulinec, P. Suquet, A computational scheme for linear and non‐linear composites
with arbitrary phase contrast, Int J Numer Methods Eng, 52 (2001) 139-160.
[30] F. Willot, Fourier-based schemes for computing the mechanical response of composites with
accurate local fields, C.R. Mécanique, 343 (2015) 232-245.
[31] T. De Geus, J. Vondřejc, J. Zeman, R. Peerlings, M. Geers, Finite strain FFT-based non-linear solvers
made simple, Comput Methods Appl Mech Eng, 318 (2017) 412-430.
[32] S. Lucarini, J. Segurado, DBFFT: A displacement based FFT approach for non-linear
homogenization of the mechanical behavior, Int J Eng Sci, 144 (2019) 103131.
[33] J. Zeman, T.W.J. de Geus, J. Vondřejc, R.H.J. Peerlings, M.G.D. Geers, A finite element perspective
on nonlinear FFT-based micromechanical simulations, Int J Numer Methods Eng, 111 (2017) 903-
926.
[34] J. Vondřejc, D.S. Liu, M. Ladecky, H.G. Matthies, FFT-based homogenisation accelerated by low-
rank tensor approximations, Comput Methods Appl Mech Eng, 364 (2020) 112890.
[35] J. Vondřejc, T.W.J. de Geus, Energy-based comparison between the Fourier-Galerkin method and
the finite element method, J Comput Appl Math, 374 (2020).
[36] S. Lucarini, M.V. Upadhyay, J. Segurado, FFT based approaches in micromechanics: fundamentals,
methods and applications, Modelling Simul Mater Sci Eng, 30 (2021) 023002.
[37] M. Kabel, D. Merkert, M. Schneider, Use of composite voxels in FFT-based homogenization,
Comput Methods Appl Mech Eng, 294 (2015) 168-188.
[38] C. Robert, C. Mareau, A comparison between different numerical methods for the modeling of
polycrystalline materials with an elastic–viscoplastic behavior, Comput Mater Sci, 103 (2015) 134-
144.
[39] J. Li, S. Meng, X. Tian, F. Song, C. Jiang, A non-local fracture model for composite laminates and
numerical simulations by using the FFT method, Compos B Eng, 43 (2012) 961-971.
27
[40] F. Willot, L. Gillibert, D. Jeulin, Microstructure-induced hotspots in the thermal and elastic
responses of granular media, Int J Solids Struct, 50 (2013) 1699-1709.
[41] F. Willot, D. Jeulin, Elastic and electrical behavior of some randommultiscale highly-contrasted
composites, Int J Multiscale Comput Eng, 9 (2011) 305-326.
[42] S. Lucarini, L. Cobian, A. Voitus, J. Segurado, Adaptation and validation of FFT methods for
homogenization of lattice based materials, Comput Methods Appl Mech Eng, 388 (2022) 114223.
[43] Q.D. To, G. Bonnet, FFT based numerical homogenization method for porous conductive materials,
Comput Methods Appl Mech Eng, 368 (2020) 113160.
[44] Z. Chen, Y.M. Xie, X. Wu, Z. Wang, Q. Li, S. Zhou, On hybrid cellular materials based on triply
periodic minimal surfaces with extreme mechanical properties, Mater Des, 183 (2019) 108109.
[45] G.I.N. Rozvany, A critical review of established methods of structural topology optimization, Struct
Multidiscipl Optim, 37 (2009) 217-237.
[46] O. Sigmund, Materials with prescribed constitutive parameters: an inverse homogenization problem,
Int J Solids Struct, 31 (1994) 2313-2329.
[47] O. Sigmund, S. Torquato, Design of materials with extreme thermal expansion using a three-phase
topology optimization method, J Mech Phys Solids, 45 (1997) 1037-1067.
[48] A.R. Diaz, O. Sigmund, A topology optimization method for design of negative permeability
metamaterials, Struct Multidiscipl Optim, 41 (2010) 163-177.
[49] S. Zhou, Q. Li, Computational design of multi-phase microstructural materials for extremal
conductivity, Comput Mater Sci, 43 (2008) 549-564.
[50] X. Huang, A. Radman, Y.M. Xie, Topological design of microstructures of cellular materials for
maximum bulk or shear modulus, Comput Mater Sci, 50 (2011) 1861-1870.
[51] J.E. Cadman, S. Zhou, Y. Chen, Q. Li, On design of multi-functional microstructural materials, J
Mater Sci, 48 (2013) 51-66.
[52] Y. Wang, Z. Luo, N. Zhang, Z. Kang, Topological shape optimization of microstructural
metamaterials using a level set method, Comput Mater Sci, 87 (2014) 178-186.
[53] R. Sivapuram, R. Picelli, Y.M. Xie, Topology optimization of binary microstructures involving
various non-volume constraints, Comput Mater Sci, 154 (2018) 405-425.
[54] M. Collet, L. Noël, M. Bruggi, P. Duysinx, Topology optimization for microstructural design under
stress constraints, Struct Multidiscipl Optim, 58 (2018) 2677-2695.
[55] J. Xu, L. Gao, M. Xiao, J. Gao, H. Li, Isogeometric topology optimization for rational design of
ultra-lightweight architected materials, Int J Mech Sci, 166 (2020).
[56] Y. Zheng, Y. Wang, X. Lu, Z. Liao, J. Qu, Evolutionary topology optimization for mechanical
metamaterials with auxetic property, Int J Mech Sci, 179 (2020) 105638.
[57] F. Dos Reis, N. Karathanasopoulos, Inverse metamaterial design combining genetic algorithms with
asymptotic homogenization schemes, Int J Solids Struct, (2022) 111702.
[58] M. Kaminski, Sensitivity analysis of homogenized characteristics for some elastic composites,
Comput Methods Appl Mech Eng, 192 (2003) 1973-2005.
[59] L. Xia, P. Breitkopf, Design of materials using topology optimization and energy-based
homogenization approach in Matlab, Struct Multidiscipl Optim, 52 (2015) 1229-1241.
[60] O. Sigmund, J. Petersson, Numerical instabilities in topology optimization: A survey on procedures
dealing with checkerboards, mesh-dependencies and local minima, Struct Optim, 16 (1998) 68-75.
28
[61] J.B. Berger, H.N.G. Wadley, R.M. McMeeking, Mechanical metamaterials at the theoretical limit of
isotropic elastic stiffness, Nature, 543 (2017) 533.
[62] P. Vogiatzis, S. Chen, X. Wang, T. Li, L. Wang, Topology optimization of multi-material negative
Poisson’s ratio metamaterials using a reconciled level set method, Comput Aided Des, 83 (2017)
15-32.
[63] J. Gao, H. Xue, L. Gao, Z. Luo, Topology optimization for auxetic metamaterials based on
isogeometric analysis, Comput Methods Appl Mech Eng, 352 (2019) 211-236.
[64] A. Radman, X. Huang, Y.M. Xie, Topological optimization for the design of microstructures of
isotropic cellular materials, Eng Optim, 45 (2013) 1331-1348.
[65] S. Sturm, S. Zhou, Y.-W. Mai, Q. Li, On stiffness of scaffolds for bone tissue engineering—a
numerical study, J Biomech, 43 (2010) 1738-1744.
